The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. • We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given Õ (√n) unentangled quantum witnesses with O (log n) qubits each. Our protocol relies on Dinur’s version of the PCP Theorem and is inherently non-relativizing. • We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. • We give evidence that QMA(2) ⊆ PSPACE, by showing that this would follow from “strong amplification” of QMA(2) protocols. • We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. • We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given Õ (√n) unentangled quantum witnesses with O (log n) qubits each. Our protocol relies on Dinur’s version of the PCP Theorem and is inherently non-relativizing. • We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. • We give evidence that QMA(2) ⊆ PSPACE, by showing that this would follow from “strong amplification” of QMA(2) protocols. • We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for kges2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. *We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O tilde(radicn) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on Dinur's version of the PCP Theorem and is inherently non-relativizing. *We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all kges=2. *We give evidence that QMA(2) sube PSPACE, by showing that this would follow from "strong amplification" of QMA(2) protocols. *We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
Stoquasticity, originating in sign-problem-free physical systems, gives rise to StoqMA, introduced by Bravyi, Bessen, and Terhal (2006), a quantum-inspired intermediate class between MA and AM. Unentanglement similarly gives rise to QMA(2), introduced by Kobayashi, Matsumoto, and Yamakami (CJTCS 2009), which generalizes QMA to two unentangled proofs and still has only the trivial NEXP upper bound. In this work, we initiate a systematic study of the power of unentanglement without destructive interference via StoqMA(2), the class of unentangled stoquastic Merlin--Arthur proof systems. Beyond its complexity-theoretic interest, StoqMA(2) is connected to the optimality of non-negative tensor optimization algorithms. We highlight:
1. NP⊆StoqMA(2) with O(n)-qubit proofs and completenes 1−2−polylog(n). Conversely, the Sum-of-Squares algorithm of Barak, Kelner, and Steurer (STOC 2014) gives an exponential-time upper bound for StoqMA(2). Our tightened analysis shows the optimality of our protocol and the BKS algorithm under ETH.
2. For logarithmic-size proofs, NP⊆StoqMA(2)log with completeness 1−O(n−2) and vanishing gap, while StoqMA(2)log⊆MA. Consequently, quantum-inspired randomness enables exponentially shorter unentangled proofs even under the assumption MA=NP.
3. For StoqMA(2)1, the parameter dependence in the general ETH-optimal time bound can be exponentially improved, or the bound achieved simultaneously with polynomial space.
Our lower bounds are obtained by stoquastizing the short-proof QMA(2) protocols using distribution testing techniques. Our upper bounds for the nearly perfect completeness case are proved via our rectangular closure testing framework.
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given ~O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
The class $\mathsf{QMA}( k) $, introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does $\mathsf{QMA}( k) =\mathsf{QMA}( 2) $ for k≥2? Can $\mathsf{QMA}( k) $ protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. We give a protocol by which a verifier can be convinced that a 3SAT formula of size m is satisfiable, with constant soundness, given $\widetilde{O}(\sqrt{m}) $ unentangled quantum witnesses with $O( \log m) $ qubits each. Our protocol relies on the existence of very short PCPs. We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any $\mathsf{QMA}( 2) $ protocol can be amplified to exponentially small error, and $\mathsf{QMA}( k) =\mathsf{QMA}( 2) $ for all k≥2. We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
We study gap amplification of the class QMA+(2) characterized by unentangled quantum proofs whose amplitudes are nonnegative in the computational basis. This class was recently introduced by Jeronimo and Wu (STOC 2023), and its behavior depends sharply on the completeness-soundness gap: although it captures the power of NEXP for some small constant gap, it is equal to QMA(2) for larger constant gap. This is in stark contrast to QMA(2) where strong gap amplification is known due to the product test by Harrow and Montanaro (FOCS 2010, JACM 2013).
In this paper, we prove for every completeness c and soundness s with c−s=1/poly(n), NEXP=QMA+(2,c,s)=QMA+(2,1−poly(n)1,41+poly(n)1). Our result gives a clean complexity phase transition for QMA+(2) since we have QMAR(2)=QMA+(2,1−poly(n)1,41−poly(n)1), where QMAR(2) denotes QMA(2) with witnesses restricted to real amplitudes. Our amplification is thus optimal in the sense that a slight improvement of our soundness would have the collapse QMAR(2)=NEXP.
Our proof combines symmetric-subspace projections with the relation QMA+(1)=NEXP of Bassirian, Fefferman, and Marwaha (ITCS 2024). The main technical ingredient is a dimension-independent de Finetti theorem in Hilbert-Schmidt norm that applies when the number of registers under consideration grows logarithmically.
This report explains the research work done during a February-June 2008 internship at MIT under the supervision of Scott Aaronson and Seth Lloyd. It exposes some necessary background knowledge in quantum mechanics, quantum computing and quantum complexity theory, and then focuses on the work conducted during this internship. The subject of this internship is to study two important classes of quantum problems, QMA and QMA(2), which consist of all languages that can be verified using respectively one and two unentangled quantum proofs. Whether these classes are equal or distinct is an open problem of great interest in quantum computing. To prove that they are the same, one can possibly simulate QMA(2) problems in QMA using a quantum operation called a disentangler. However, it has been conjectured that polynomial disentanglers do not exist, and therefore this approach fails. In this report, we investigate this conjecture and give two results: in a specific situation, when exponential precision is required, this conjecture holds as long as P 6 = NP. Moreover, in the same situation, we show that the conjecture could be proven unconditionally using a stronger hypothesis.